Conjugacy functor whose normalizer generates whole group with p'-core controls fusion
Statement
Suppose is a finite group, is a prime number, and is a -conjugacy functor on whose normalizer, along with the p'-core, generates the whole group. Explicitly, this means that if is a -Sylow subgroup of :
.
Then, controls fusion in . In other words, any two subsets of that are conjugate in are also conjugate in .
References
- Finite Groups by Daniel Gorenstein, ISBN 0821843427, Page 282, Theorem 4.1, (Section 8.4)^{More info}